Classical' Age-Modelling of Radiocarbon Sequences

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Quaternary Geochronology

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Research Paper Methods and code for ‘classical’ age-modelling of radiocarbon sequences

Maarten Blaauw*

School of Geography, Archaeology and Palaeoecology, Queen's University Belfast, United Kingdom article info abstract

Article history: Ageedepth models form the backbone of most palaeoenvironmental studies. However, procedures for Received 19 February 2009 constructing chronologies vary between studies, they are usually not explained sufﬁciently, and some are Received in revised form inadequate for handling calibrated radiocarbon dates. An alternative method based on importance 22 October 2009 sampling through calibrated dates is proposed. Dedicated R code is presented which works with calibrated Accepted 27 January 2010 radiocarbon as well as other dates, and provides a simple, systematic, transparent, documented and Available online xxx customizable alternative. The code automatically produces ageedepth models, enabling exploration of the impacts of different assumptions (e.g., model type, hiatuses, age offsets, outliers, and extrapolation). Keywords: Ó Radiocarbon age-modelling 2010 Elsevier B.V. All rights reserved. Calibration Software Exploratory analysis

1. Introduction “classical” ageedepth modelling such as linear interpolation (31) or linear/polynomial regression (18). Ageedepth models are built to estimate the calendar ages of As will be discussed below, calibrated 14C dates often have irreg- depths in a core from a deposit, based on limited numbers of dated ular distributions which cannot be properly reduced to single point depths and on assumptions as to how the deposit has accumulated estimates with symmetric error bars. However, most age-modelling between those dated depths. Given the intensive use of ageedepth routines do indeed simplify calibrated distributions to symmetric modelling, surprisingly few details have been published about the ones. Bennett (1994) and Bennett and Fuller (2002) discussed basic techniques involved, adequate software seems to be largely missing ageedepth models based on linear interpolation, polynomial and or under-used, and details of applied ageedepth modelling spline regression, but based on uncalibrated radiocarbon dates. Proxy assumptions are often missing in the literature. graphing software Tilia (Grimm, 1990)producesageedepth models, A review of all papers published in 2008 in the journals Quater- but i) does not calibrate dates, ii) does not take into account errors, nary Geochronology, Quaternary Science Reviews, Journal of and iii)doesnotprovideageedepth graphs. Heegaard et al. (2005) Quaternary Science and The Holocene, indicates 93 papers applying applied mixed-effects modelling, but assumed that calibrated 14C some kind of ageedepth model through fossil proxy deposits ages can be approximated by a simpler statistical distribution (e.g., (Table 1). Of these papers, all 82 that used 14C dates provide details Gaussian). It is likely that studies which do not report their about their calibration procedures (e.g., which calibration curve and ageedepth modelling software often use regression or spline func- software were applied). However, often much less information is tions which generally cannot deal with irregular distributions. supplied about subsequent ageedepth modelling (regularly this Whereas proxy graphic software psimpoll (Bennett,1994) can handle information can only be gleaned from graphs). Sixty papers do not tell calibrated 14C dates, they need to be calibrated outside the software. which (if any) point estimates were used, 71 do not mention the Although Bayesian methods can produce very reliable ageedepth software used to produce ageedepth curves, 65 do not indicate models (e.g., Blaauw and Christen, 2005; Bronk Ramsey, 2008), whether ageedepth uncertainties were calculated, and 18 do not Table 1 shows that by far most studies continue to use more basic, specify the applied ageedepth model. Eleven papers apply some ‘classical’ age-depth models (especially for sites with only a handful form of Bayesian ageedepth modelling, whereas the remainder use of dates). Here these classical approaches will be discussed, and a technique based on probability sampling methods will be proposed in order to take into account the multi-modal and asymmetric nature of calibrated 14C dates. A more systematic and transparent approach e * Tel.: þ44 28 9097 3895. to produce classical age depth models is presented, based on free E-mail address: [email protected] and open-source software.

Please cite this article in press as: Blaauw, M., Methods and code for ‘classical’ age-modelling of radiocarbon sequences, Quaternary Geochronology (2010), doi:10.1016/j.quageo.2010.01.002 ARTICLE IN PRESS

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Table 1 Literature analysis of primary literature reporting ageedepth models, published in 2008 in Quaternary Geochronology (2 papers), Quaternary Science Reviews (40), Quaternary Research (10), Journal of Quaternary Science (13) and The Holocene (28). Publications citing previously published ageedepth models were not taken into account. As several papers applied a number of ageedepth models and types of dates, the numbers do not always add up. 17 papers mentioned the removal of dates identiﬁed as outlying.

Dates Point estimate Model Model error Age-model software 14C (82) Not specified (60) Linear interpolation (31) Not specified (65) Not specified (71) Tephra (11) Full distribution (13) Not specified (18) 2 sd error (17) Oxcal (6) 210Pb/Cs (9) Mid (5) Linear regression (13) 1 sd error (6) Bpeat (4) U/Th (8) Median (4) Bayesian (11) Mixed-effect (3) OSL (5) Intercept (3) Linear regression (5) Bchron (1) Tuning (4) Mean (1) Spline (4) psimpoll þ BCal (1) Varves (2) Weighted mean (1) Mixed-effect (3) Other (2) Mid of most probable range (1) CRS (2) Other (2)

2. Radiocarbon calibration yjqwN mðqÞ; s2 (1)

14 The importance of using calibrated C dates cannot be overstated. where j means ‘given’,ands is a combination of the measure- 14 Owing to past changes in atmospheric Cconcentration,therela- ment uncertaintyqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and that of the calibration curve at the calendar 14 tionship between C and calendar age is far from linear, thus age, s(q), s ¼ sd2 þ sðqÞ2. Usually the total probability over a large requiring a calibration curve (Reimer et al., 2009). Whereas an range of calendar years is normalised to 100%. Because the resolution 14 uncalibrated C date can be assumed to have symmetrical errors of most calibration curves is decadal, calibration at yearly intervals (normal distribution), calibration will often result in widened, requires (usually linear) interpolation of m(q) and s(q). asymmetrical and multi-peaked calendar age uncertainties (Fig.1;all The currently internationally accepted calibration curves are calendar ages are expressed in years before AD 1950 or cal BP, unless IntCal09 (northern hemisphere atmospheric; Reimer et al., 2009), mentioned otherwise). These asymmetrical distributions can cause Marine09 (for marine dates; Reimer et al., 2009), and SHCal04 problems when applying classical interpolation/regression tech- (southern hemisphere atmospheric; McCormac et al., 2004). niques, as they often assume symmetrically/normally distributed Alternative curves exist, e.g., for dating organic material deposited errors. after nuclear weapons testing (Hua and Barbetti, 2004; Levin et al., 14 Standard C calibration software generally applies ‘probabilistic 2008). 14 calibration’ to provide calendar age estimates of C dates (Stuiver In case any of the 14C dates in a sequence is suspected to be prone and Reimer, 1989, 1993; van der Plicht and Mook, 1989). In this to a systematic age-offset (e.g., owing to a lake hard-water effect, or 14 method, a C date with reported mean and error y sd, is assumed to an additional local reservoir effect for marine data; see Reimer and 14 to have a Gaussian distribution on the C age scale; y w N(y, sd). Reimer, 2001, http://calib.org/marine/), this should be corrected for 14 Whereas fluctuations in the calibration curve often cause single C before calibrating. If an uncertainty estimate of the reservoir effect is ages to correspond to multiple calendar ages, every calendar age 14 available,qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the error of the C date should be increased accordingly will always have only one single corresponding 14C age. Therefore, (sd ¼ sd2 þ sd2 ,wheresd is the error of the reservoir effect). to calibrate a 14C date, we aim to find the probability of that date res m res belonging to a given calendar year. For a range of calendar years q, Calibrated distributions are usually reduced to 1 or 2 sd calibrated the corresponding 14C age of the calibration curve, m(q), is looked up ranges, taking into account their asymmetric and multi-peaked and compared with the 14C measurement (see Christen, 1994): shapes. In standard calibration software such as CALIB (Stuiver et al., 2005), these ranges are obtained by calculating the highest posterior density (hpd) ranges: i) probability distributions are normalised to 100% (Eq. (1)), ii) the calendar years are ranked according to their probabilities, iii) those calendar ages with a cumulative summed probability at or below the 1 sd (c. 68%) or 2 sd (c. 95%) threshold are retained, and iv) the extremes and probabilities of any sub-ranges within these calendar ages are found (Fig. 1). The probabilities within hpd ranges are either reported as ratios of the entire calibrated distribution (where the hpd ranges add up to say 68% or 95%; OxCal, Bronk Ramsey, 2008), or as ratios of the hpd region only (where the ranges are normalised to 100%, e.g., CALIB, Stuiver et al., 2005). Calibrated ranges at 1 sd will obviously result in narrower confidence intervals, and thus a perceived higher precision, than at 2 sd. However, given the often asymmetric and multi-modal nature of calibrated distributions, the probability that the ‘true’ calendar date lies outside the 1 sd ranges is considerable (c. 32%). Therefore the use of non-normalised 2 sd calibrated ranges is preferable. 14 Fig. 1. Calibration of a 14C date of 2450 30 14C BP at 2 standard deviation. Highest Non- C dating information, such as a tephra layer of known or posterior density ranges (dark grey) are 2359e2546 (probability 57.6%), 2560e2574 estimated calendar age, a historically anchored pollen event, or (2.5%), 2578e2616 (10.6%) and 2635e2701 cal BP (24.7%). Axes of the probability a modern age for the top of a sequence (e.g., the year of sampling), distributions are of arbitrary heights. From left to right, white crosses indicate the can provide additional dating points for age-depth modelling. date's midpoint, weighted mean, median and mode, respectively. Multiple calendar ages intercept 2450 14C BP; the chosen intercept had the highest probability when Whenever available, it is advisable to include such data, with an taking the error on the calibration curve into account. estimate of their chronological uncertainty.

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3. Ageedepth modelling 0 Having estimated the age distributions of the individual dated depths, the next step is to provide age estimates for all depths in a sequence. This is done by modelling the accumulation of the 001 sequence through time. The researcher will have to decide on the )mc(htpeD most likely type of ageedepth model. For example, a deposit in a stable environment will probably have accumulated with fewer 002 events of hiatuses or accumulation rate changes than one from a more variable environment, and should thus probably be modelled using a smooth age-depth model. The very common ageedepth 003 model of linear interpolation between the dated levels (Table 1), assumes that abrupt changes in accumulation rate took place exactly at the dated depths. Although this assumption is often likely to be wrong, linear interpolation often produces what look like plausible 2500 2000 1500 1000 500 0 ageedepth models (Bennett, 1994). In some sequences one or more cal BP hiatuses appear present, requiring models through subsets of the e data. Additionally, at times ageedepth models need to be extrapo- Fig. 3. Schematic drawing of an iterative sampling process in age depth modelling. Four point estimates (coloured crosses) are sampled from calibrated distributions lated beyond the dated levels (necessarily introducing a large degree (grey histograms), and error-weighted smooth splines drawn through the crosses of age-model uncertainty for the undated parts of the sequence). (each colour corresponds to an iteration process). Note that the splines do not necessarily Calibrated distributions and ranges of 14C dates, and resulting cross each individual age point estimate. Splines were weighted by the calibrated prob- uncertainties in ageedepth models, often span several centuries. abilities of the sampled calendar ages. The grey envelope shows the final 95% confidence intervals based on 1000 iterations. The age distribution of all splines at 200 cm depth is Even so, users often require a single ‘best’ ageedepth model in order indicated by the blue histogram. The shown data are a selection from core Quilichao to plot proxy values against calendar age. Drawing straight, poly- (Berrio et al., 2002; Fig. 4). (For interpretation of the references to colour in this figure nomial, or exponential curves through ageedepth data with legend, the reader is referred to the web version of this article.) symmetrical errors is straightforward and can be done automatically in spreadsheets and age-modelling software such as psimpoll (Bennett, 2008), the ageedepth code of Heegaard et al. (2005),and which depend on symmetric error distributions should not be Tilia (Grimm, 1990). Similarly, confidence intervals of such interpo- applied to 14C ageedepth modelling. lation or regression curves can be easily calculated using standard A proposed alternative to the above point estimates is to use the techniques. entire calibrated distribution of the dates. This can be done by However, 14C ages are notorious for their highly asymmetric repeated random sampling of the calibrated distributions of the calibrated distributions, and reducing them to single points is dates, each time calculating an ageedepth model through the problematic (Telford et al., 2004b; Michczynski, 2007). To take the sampled ages. If a large number of calendar ages is drawn from date of Fig. 1 as example, its midpoint of the 95% hpd ranges lies at a calibrated distribution f(q), with the probability of an age being 2530 cal BP, a year with rather low calibrated probability. Also other sampled proportional to its height of the probability distribution at point estimates such as the median (2508 cal BP) or the weighted that age (see Eq. (1)), a histogram of these age estimates will form mean (2528 cal BP) fall at unlikely calendar ages. The year with the a close approximation of the original calibrated distribution (Fig. 2). highest probability (mode or intercept; 2486 cal BP) is an inadvis- This technique is known as importance, bootstrap or Monte Carlo able point estimate, owing to i) the possibility of multiple inter- sampling; Bayesian ageedepth modelling makes use of comparable secting calendar years and ii) its sensitivity to small variations in the though more sophisticated iterative sampling methods. calibration curve (Telford et al., 2004b). Moreover, the error ranges Similar to estimating calibrated distributions by importance of all these point estimates (except the midpoint) will often be sampling, for a core with multiple dates we can sample ages from asymmetric. Therefore classical statistical regression techniques each of its calibrated distributions (fn(q) for each of n dated depths), thus resulting in point age estimates for all dated depths (Fig. 3). A curve can then be drawn through these ageedepth point estimates, resulting in a calendar age estimate for any depth d (dated or undated). Repeated sampling and subsequent ageedepth modelling 0 400. will thus result in multiple age estimates for depth d,andhistograms from a sufficiently large amount of iterations will provide us with a calendar age distribution of this depth, fd(q). The explicit assump-

ytisneD tion made here is that although the true calendar ages of a core's 200.0 dated depths will never be known, the type of ageedepth curve applied through these unknown points (e.g., a smooth spline) will be a reliable approximation of the unknown true accumulation history of the site. Through repeated sampling of the calibrated distributions, 0.0 00 each time calculating its ageedepth model, we will thus build up an 2800 2700 2600 2500 2400 2300 approximation of the site's accumulation history, based on the dated depths, their uncertainties, and a chosen type of age-model. This cal BP procedure can be run in psimpoll (Bennett, 2008) by feeding it age distributions calculated by, e.g., BCal (Buck et al., 1999; for an appli- Fig. 2. Two thousand point calendar year estimates (short lines at bottom graph) were cation see Yeloff et al., 2006). A comparable bootstrap procedure was sampled from the calibrated distribution of the date of Fig. 1 (dark grey distribution). reported by Higuera et al. (2009). As explained in the text, calendar years with higher probabilities were sampled more frequently. A histogram of the sampled ages (light grey bars) closely resembles the Heegaard et al. (2005) using mixed-effect modelling to add an original calibrated distribution. extra error term to dated levels, with the aim to include an

4 M. Blaauw / Quaternary Geochronology xxx (2010) 1e7 uncertainty as to whether the reported date of a dated level is a true individually through calculating their calibrated distributions for representation of its real age (e.g., see Scott, 2007 for the variability every calendar year as in Eq. (1) (default in cal BP; cal BC/AD as between repeated 14C measurements on a single piece of wood). alternative). In case a 14C date is suspected to be outlying, it can be However, Heegaard et al. (2005) did not use calibrated distributions removed manually and is then not taken into account in the age- and instead assumed that these can be approximated by symmet- model. Any 14C dates which fall outside the ranges of the calibration rical distributions. For calibrated distributions, the “Heegaard” curve will not be used, while dates which fall partly beyond its ranges added error could be simulated by sampling say 1000 14C ages from will be truncated (in both cases a warning will be given). Highest a reported 14C age (y sd), assuming a normal distribution for the posterior density probabilities are not normalised (this differs from 14C measurement. Then each sampled 14C age is calibrated, and e.g., CALIB; see Section 2). Depending on the core and chosen settings, a single calendar year is sampled from its calibrated distribution (Eq. calculations take up to c. 10e20 s on a modern PC. (1)). As with the sampling method described above, a histogram of Ageedepth models are constructed using the calibrated distri- the mixed-effect sampled calendar ages will form an estimate of the butions described above. Several common types of age-models are date's calibrated distribution. As expected, mixed-effect age-models provided here; i) linear interpolation between the dated levels, generally obtain wider confidence limits than models without the ii) linear or higher order polynomial regression, and iii) cubic, added error (data not shown). smoothed or locally weighted splines (the latter two can optionally Even if all uncertainties have been taken into account, users often be weighted by the inverse and squared dating errors). Sometimes need a single ‘best’ ageedepth model in order to produce plots of the the stratigraphy or the 14C dates indicate the presence of hiatuses core's analysed proxies against age (however, see Blaauw et al., 2007 (Blaauw and Christen, 2005); clam comes with an option to include for plots which include chronological uncertainties of proxies). As one or more hiatuses. Extrapolation beyond the 14C dated levels is explained above, there are considerable problems with age-depth inadvisable but can be done if required. models through point estimates based on the midpoints, Confidence intervals for the undated levels are calculated using (weighted) mean, median or mode of calibrated dates. Even more, the the Monte Carlo approach detailed above: i) From the calibrated considerable scatter between closely spaced dates, as well as age distribution of every dated depth a calendar age is sampled between dates and the applied model, indicates that the accuracy of (with the probability of a year being chosen proportional to its 14C measurements should not be overestimated (see Scott, 2007 for calibrated probability p(q) as calculated from Eq. (1)); ii) an age- the variable outcome of repeated 14C dating of a single sample). As an model is drawn through these points, and iii) for every depth alternative to drawing ‘best’ ageedepth models through point esti- (default 1 cm steps) the resulting calendar age is stored. Per default, mates from individual dates, I propose to use the point ageedepth curves with age-reversals are rejected with a warning. In step i,an estimates derived from the ageedepth model itself, as it is based on extra error term can be included akin to the mixed-effect models all dating information and contains an interpretation of a likely described in Heegaard et al. (2005; here the size of the extra error is accumulation history for a site. A robust point estimate for any depth taken to be equal to a date's reported error). The sampling process in a core could then be provided by the weighted mean of all age- is repeated often (default 1000 times, which suffices for most initial modeliterationsatthatdepth. runs; use e.g., 10 times more iterations for final runs), resulting in as many calendar age estimates for every depth. For every depth, its 4. Alternative age-modelling software calendar age point estimates are binned, ranked, normalised and summed, after which the range of those binned calendar ages Here software is presented which performs the methods detailed which fall within e.g., 68% (1s) or 95% (2s) forms an estimate of the above simultaneously and in an automated and documented manner. confidence interval for every individual depth. As the calculations The approach can be used to test the results of different age-model are based on random iterations, a slightly different confidence choices, and to systematically and rapidly produce publication- interval will be estimated for every run. quality graphs in addition to files containing information about the To find the single ‘best’ ageedepth model from all iterated settings, results, and quality of the obtained age-model. The code ageedepth models, clam calculates for every depth the weighted produces non-Bayesian, ‘classical’ ageedepth models, and therefore mean of all sampled calendar ages (default), or alternatively, the is called clam. It can also be used to calibrate single 14C dates. midpoint of the highest posterior density range of all calendar ages Clam has been written in the open-source statistical environ- (default at 95% confidence intervals). In order to facilitate compari- ment R (R Development Core Team, 2010). R can be downloaded sons with commonly used although inadvisable point estimates freely at http://www.r-project.org for Windows, Mac and Linux (Table 1), clam can provide ageedepth models based on i) the modes (current version 2.10.1, c. 25 MB). The software makes use of or intercepts (highest p(q) of Eq. (1)), ii)themedians,iii)theweighted a command-window, which may not appear very user-friendly at means, or iv) the midpoints of the highest posterior densities. The first sight. However, for clam the number of commands to be typed product of the probabilities p(q) (see Eq. (1)) of the calendar age is limited. Moreover, this approach enables a highly customizable estimates of the dated levels provides us with a measure of the environment. R is relatively easy to learn and is being used for model's goodness-of-fit. If age-reversals occur in the chosen age- a growing number of applications in palaeo-research (Telford et al., model, a warning is given. 2004b; Blaauw and Christen, 2005; Heegaard et al., 2005; Blaauw et al., 2007, 2010; Haslett and Parnell, 2008). R is similar to S and 5. Case studies clam could thus also be used in S-PLUS (Insightful Corp.). Clam comes with a manual which contains information for installing and Telford et al. (2004a) tested the accuracy of ageedepth modelling using the code (see Section 8). techniques by modelling the 14C dates of a sediment sequence dated In short, clam calibrates the 14C dates (calendar scale dates such as independently through varve counting. Here a slightly different tephras of known calendar age can be used as well), constructs an approach is taken, simulating the accumulation of sediment over age-model based on decisions made by the user, and produces graphs time and its subsequent 14C dating and ageedepth modelling (Fig. 4). and text files. Provided calibration curves are IntCal09, Marine09 Sediment starts to accumulate at a rate of 20 yr/cm for the upper (both Reimer et al., 2009) and SHCal04 (McCormac et al., 2004), while depth. Each depth further down has an accumulation rate differing other curves such as post-bomb curves can be added. After correcting slightly from its overlying depth, by sampling from a normal distri- for any reservoir effect, all 14C dates of a sequence are calibrated bution with the previous accumulation rate as mean. The standard

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m 01 0 c( h 140 tpeD D

160 150 1 05

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6000 5000 4000 3000 2000 12000 8000 6000 4000 2000 12000 8000 4000 0 cal BP cal BP cal BP

Fig. 4. Simulated accumulation (red curve; see text) and 14C dating (blue blocks show 95% hpd ranges) of a sediment record, followed by clam ageedepth modelling using a smooth spline (error-weighted with smoothness set at the default of 0.3; grey envelopes show 95% confidence intervals). The first two simulations used 7 dates, while the last one has 14 dated levels. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

deviation of this distribution dictates the variability in accumulation 6. Discussion rate (here set at 7). The cumulative sum of all simulated accumulation rates then forms the accumulation history of the core. Next, a number The software described here does not pretend to offer the latest of depths are sampled randomly from all depths (using a uniform advances in 14C modelling. Rather it attempts to provide an easy, distribution), each of which stems from a known “true” calendar year. automated, transparent, documented and adaptable environment for For each of these depths, the 14C year corresponding to its calendar producing age-models from 14C sequences. Indeed, given the current year is found in the IntCal09 calibration curve (Reimer et al., 2009). suboptimal approaches and lack of information in most published After adding random noise to this 14C age by sampling from a normal ageedepth models (Table 1), it is hoped that clam could aid in distribution with the 14C age as mean and 30 year as standard devi- standardising and enhancing future ‘classical’ ageedepth models. ation, this value serves as a ‘de-calibrated’ 14Cage(Telford et al., Other software exists for producing age-models of 14Cdated 2004a). Finally clam calculates an age-depth model through the 14C sequences. In psimpoll (Bennett, 2008) one can choose from a large dates and their depths (Fig. 4). number of age-models, but 14C dates are left uncalibrated or need to Generally, the ageedepth models produced by clam seem to be calibrated outside the program (e.g., in BCal; Buck et al.,1999), and provide reasonable approximations of the true (simulated) accu- hiatuses cannot be included. Tilia (Grimm, 1990) is popular for mulation history. Occasionally the true curve lies outside the calcu- ageedepth model construction, but cannot handle hiatuses or lated conﬁdence intervals (Fig. 4b), but these deviations tend to occur asymmetric dates, and does not take into account the errors of dates. at sections of low dating density. With more dates, the ageedepth Heegaard et al. (2005) provide R code for 14C age-modelling based on models tend to follow the true history more closely (Fig. 4c). mixed-effect regression, but there calibrated 14C dates are assumed to The effects of different age-modelling choices can be checked have a normal distribution on the calendar scale, hiatuses are not from a case-study; core Quilichao from a lake in Colombia (Berrio taken into account, and only one type of age-model can be chosen. et al., 2002). This core with 9 14C dates, and an assumed age of AD Blaauw and Christen (2005) published Bayesian software Bpeat for 2000 5 for its surface, could be age-modelled assuming linear ‘wiggle-matching’ 14C dates based on the age-model of piece-wise interpolation together with extrapolation beyond the dated levels to linear accumulation; other age-models are not included in their 800 cm (Fig. 5a). Between 485 and 445 cm depth, a large jump in 14C program. OxCal (Bronk Ramsey, 2008)providesanumberof ages appears, suggesting a hiatus. In Fig. 5b, the age-model is linear ageedepth models based on Bayesian statistics, and recently regression, assuming a hiatus at 477 cm depth (the level where Bayesian software based on a monotonous accumulation model has pollen spectra change abruptly; Berrio et al., 2002). In Fig. 5c, the been published (Haslett and Parnell, 2008). same hiatus is assumed, and the model is based on a smooth spline. The Bayesian age-modelling routines mentioned in the previous Fig. 5d, a smooth spline is drawn through all dates except the one at paragraph (Blaauw and Christen, 2005; Bronk Ramsey, 2008; 445 cm which is considered an outlier. Haslett and Parnell, 2008) use advanced, robust and ﬂexible The age-models in this example were chosen rather arbitrarily numerical methods, and should be the preferred approach for and for illustration purposes only (although the reader is reminded future ageedepth models (Blockley et al., 2007). However, we have that at least some ageedepth models in the recent literature appear seen that many studies continue to apply ‘classical’ ageedepth to have been chosen in a similarly arbitrary way; see Table 1). Users modelling routines, and existing Bayesian ageedepth modelling should apply those settings which are deemed adequate for the methods might not add much extra value to low-resolution dated archive in question (such as deciding between linear interpolation sites. Therefore, the intended use of clam is to rapidly, transparently and polynomial or spline regression as the most likely model given and systematically produce age-models, while exploring the impact the site's deposition regime and/or stratigraphy). of different modelling choices (e.g., type of age-model, presence or

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20000 15000 10000 5000 0 20000 15000 10000 5000 0 cal BP cal BP

Fig. 5. Age-models of core Quilichao (Berrio et al., 2002). The age-models are extrapolated down to 800 cm. a, age-model based on linear interpolation between the dates levels. Between 485 and 445 cm depth, a large jump in 14C ages appears, suggesting a hiatus. b, linear regression age-model, assuming a hiatus (horizontal dashed grey line) at 477 cm depth. c, the same hiatus is assumed, now using an error-weighted smooth spline (default smoothing parameter 0.3) as age-model. The core surface was assumed to be of recent age (AD 2000 5). d, as in c but without hiatus and regarding the date at 445 cm date as outlying. Colours as in Fig. 4. absence of a hiatus, reservoir effect, outliers, or anchor points). As and Maytee Segura for stimulating discussions about age- a next step in the age-modelling process, more sophisticated modelling. Heather Binney and many more researchers and approaches relevant for the core in question should be considered students are thanked for reporting bugs and desired features while (Blockley et al., 2007). testing clam on their cores. It is vital to note the dangers of reducing calibrated 14C dates to single-year point estimates and subsequent drawing of a single Appendix. Supplementary information ageedepth curve through these points. Point age estimates of calibrated 14C dates and undated levels should not be regarded as Supplementary information associated with this article can be the truth (e.g., see Telford et al., 2004b; Michczynski, 2007; Blaauw found, in the online version, at doi:10.1016/j.quageo.2010.01.002 et al., 2007; Scott, 2007). Rather much attention should be paid to and http://chrono.qub.ac.uk/blaauw/. the estimated uncertainties for both the calibrated dates and the entire sequence. References The R code of clam can be understood relatively easily. Indeed, users are invited to adapt the code to their preferences, such as the Bennett, K.D., 1994. Conﬁdence intervals for age estimates and deposition times in e ﬁgure settings. The open nature of the code enables the interested late-Quaternary sediment sequences. The Holocene 4, 337 348. Bennett, K.D., Fuller, J.L., 2002. Determining the age of the mid-Holocene Tsuga can- user to open the ‘black box’ of age-modelling, following in detail adensis (hemlock) decline, eastern North America. The Holocene 12, 421e429. how 14C dates can be calibrated and age-models derived. With this Bennett, K.D., 2008. Psimpoll v. 4.263. http://chrono.qub.ac.uk/psimpoll/psimpoll.html in mind, clam is also of use for educational purposes. (last accessed 17.02.2009.). Berrio, J.C., Hooghiemstra, H., Marchant, R., Rangel, O., 2002. Late-glacial and Holocene history of the dry forest area in the south Colombian Cauca Valley. Journal of Quaternary Science 17, 667e682. Acknowledgements Blaauw, M., Christen, J.A., 2005. Radiocarbon peat chronologies and environmental change. Applied Statistics 54, 805e816. Blaauw, M., Christen, J.A., Mauquoy, D., van der Plicht, J., Bennett, K.D., 2007. Testing The ideas for clam were born during a stay at the Centro de the timing of radiocarbon-dated events between proxy archives. The Holocene Investigación en Matemáticas (CIMAT), Mexico. I thank Andrés 17, 283e288. Christen for introducing me to R, Juan Carlos Berrio for providing Blaauw, M., Wohlfarth, B., Christen, J.A., Ampel, L., Veres, D., Hughen, K.A., Preusser, F., Svensson, A., 2010. Were last glacial climate events simultaneous the Quilichao data, and Bas van Geel, Hans van der Plicht, Dmitri between Greenland and France? A quantitative comparison using non-tuned Mauquoy, Paula and Ron Reimer, Keith Bennett, Andrés Christen chronologies. Journal of Quaternary Science 25, 387e394.

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Please cite this article in press as: Blaauw, M., Methods and code for ‘classical’ age-modelling of radiocarbon sequences, Quaternary Geochronology (2010), doi:10.1016/j.quageo.2010.01.002